Application Of Partial Differential Equation In Digital Image Processing

With the development of the era, the existence value of digital image is becoming more and more obvious. Both daily life and scientific research, the appearance of digital image, brings us great convenience, even changes the way of life in today. However, how to get the digital image of high quality is an important problem that people will face. In order to improve image quality which generally is not high, scholars and experts n this field according to the characteristics of the original image problems, mainly summarizes the probability and statistics, the wavelet analysis and partial differential equation of three ways to improve it. This thesis was based on the perspective of partial differential equations to study the problems in the digital image processing, mainly in the noise removal, inpainting and blur removal. According to the digital images stored in the computer. This thesis was consisted of the gray image and color image two aspects, respectively study above the main problems in the digital image. Image in mathematics is expressed as a binary scalar or vector function, so the image processing ;an be regarded as a search for a specific binary function under the condition of problem. From mathematics essence, it is a dual variational problem under the limited conditions. We can solve it in this way, Firstly by the method of Lagrange Multiplier to construct a new object functional, Secondly according to the variational principle, the solution of the problem will be obtained to meet the Euler-Lagrange equation. Then using projection gradient method to construct the corresponding gradient flow equation, at the same time theoretically proved that:by the time evolution of the gradient flow ;an convergent to the original solution of variational problem. The finite difference method is used to discrete the gradient flow equation, the steady-state solution of the gradient flow equations will be obtained. Steady-state solution will be as the final result of image processing. Due to the diversity of the variational problem of functional selection, this thesis put forward two kinds of different models, heat equation model and the total variation model, and also simply introduce about the deformation model of total variation model-C-D-D model. Finally we summed up the general theoretical model, which provides a theoretical support for further deeper image processing. By detailed numerical examples analysis, we point out that the proposed model in digital image processing is roughly coincide with theoretical analysis, the result of the partial differential equation method in digital image denoising, inpainting, and deblurring and so on can play a certain effect, can effectively change the present situation of the specific digital image quality which is not high.